How to make a rigged object bend when pressed against a surface?
The best way would be to use Blender's Lattice tool.

https://docs.blender.org/manual/en/dev/modeling/modifiers/deform/lattice.html

it allows you to use a lower resolution 'mesh cage' to deform your model. there are two tutorials on the blender wiki, and this youtube link below is a good tutorial.

https://www.youtube.com/watch?v=10asQDMhFQw